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88
Problems in Computational Geometry
 Packing and Covering
, 1974
"...  reproduced, stored In a retrieval system, or transmlt'ted, In any form or by any means, electronic, mechanical, photocopying, or otherwise, without the prior written permission of the author. ..."
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Cited by 454 (2 self)
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 reproduced, stored In a retrieval system, or transmlt'ted, In any form or by any means, electronic, mechanical, photocopying, or otherwise, without the prior written permission of the author.
Facility location models for distribution system design
, 2004
"... The design of the distribution system is a strategic issue for almost every company. The problem of locating facilities and allocating customers covers the core topics of distribution system design. Model formulations and solution algorithms which address the issue vary widely in terms of fundamenta ..."
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Cited by 33 (0 self)
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The design of the distribution system is a strategic issue for almost every company. The problem of locating facilities and allocating customers covers the core topics of distribution system design. Model formulations and solution algorithms which address the issue vary widely in terms of fundamental assumptions, mathematical complexity and computational performance. This paper reviews some of the contributions to the current stateoftheart. In particular, continuous location models, network location models, mixedinteger programming models, and applications are summarized.
An efficient algorithm for minimizing a sum of Euclidean norms with applications
 SIAM Journal on Optimization
, 1997
"... Abstract. In recent years rich theories on polynomialtime interiorpoint algorithms have been developed. These theories and algorithms can be applied to many nonlinear optimization problems to yield better complexity results for various applications. In this paper, the problem of minimizing a sum o ..."
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Cited by 23 (4 self)
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Abstract. In recent years rich theories on polynomialtime interiorpoint algorithms have been developed. These theories and algorithms can be applied to many nonlinear optimization problems to yield better complexity results for various applications. In this paper, the problem of minimizing a sum of Euclidean norms is studied. This problem is convex but not everywhere differentiable. By transforming the problem into a standard convex programming problem in conic form, we show that an ɛoptimal solution can be computed efficiently using interiorpoint algorithms. As applications to this problem, polynomialtime algorithms are derived for the Euclidean single facility location problem, the Euclidean multifacility location problem, and the shortest network under a given tree topology. In particular, by solving the Newton equation in linear time using Gaussian elimination on leaves of a tree, we present an algorithm which computes an ɛoptimal solution to the shortest network under a given full Steiner topology interconnecting N regular points, in O(N √ N(log(¯c/ɛ)+ log N)) arithmetic operations where ¯c is the largest pairwise distance among the given points. The previous bestknown result on this problem is a graphical algorithm which requires O(N 2) arithmetic operations under certain conditions. Key words. polynomial time, interiorpoint algorithm, minimizing a sum of Euclidean norms, Euclidean facilities location, shortest networks, Steiner minimum trees
Analyzing and modeling the maximum diversity problem by zeroone programming. Decision Sci
, 1993
"... The problem of maximizing diversity deals with selecting a set of elements from some larger collection such that the selected elements exhibit the greatest variety of characteristics. A new model is proposed in which the concept of diversity is quantifiable and measurable. A quadratic zeroone model ..."
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Cited by 19 (2 self)
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The problem of maximizing diversity deals with selecting a set of elements from some larger collection such that the selected elements exhibit the greatest variety of characteristics. A new model is proposed in which the concept of diversity is quantifiable and measurable. A quadratic zeroone model is formulated for diversity maximization. Based upon the formulation, it is shown that the maximum diversity problem is NPhard. 'Tho equivalent linear integer programs are then presented that offer progressively greater computational efficiency. Another formulation is also introduced which involves a different diversity objective. An example is given to illustrate how additional considerations can be incorporated into the maximum diversity model. Subject Areas: Discnk hgmmming, Linear Rvgmmming, and Mathematical hgmmming.
Computational Geometry and Facility Location
 Proc. International Conference on Operations Research and Management Science
, 1990
"... this paper we briefly survey the most recent results in the area of facility location, concentrating on versions of the problem that are likely to be unfamiliar to the transportation and management science community and we explore the interaction between facility location problems and the field of c ..."
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Cited by 18 (3 self)
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this paper we briefly survey the most recent results in the area of facility location, concentrating on versions of the problem that are likely to be unfamiliar to the transportation and management science community and we explore the interaction between facility location problems and the field of computational geometry. Such versions of the problem include the standard models of points as customers and facilities but with geodesic rather than the traditional Minkowski metrics as measures of distance, as well as more complicated models of customers and facilities such as
A Fast and Robust General Purpose Clustering Algorithm
 In Pacific Rim International Conference on Artificial Intelligence
, 2000
"... General purpose and highly applicable clustering methods are usually required during the early stages of knowledge discovery exercises. kMeans has been adopted as the prototype of iterative modelbased clustering because of its speed, simplicity and capability to work within the format of very larg ..."
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Cited by 16 (2 self)
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General purpose and highly applicable clustering methods are usually required during the early stages of knowledge discovery exercises. kMeans has been adopted as the prototype of iterative modelbased clustering because of its speed, simplicity and capability to work within the format of very large databases. However, kMeans has several disadvantages derived from its statistical simplicity. We propose an algorithm that remains very efficient, generally applicable, multidimensional but is more robust to noise and outliers. We achieve this by using the discrete median rather than the mean as the estimator of the center of a cluster. Comparison with kMeans, Expectation Maximization and Gibbs sampling demonstrates the advantages of our algorithm.
Optimisation for Surface Mount Placement Machines
 Proc. of the IEEE ICIT’02
, 2002
"... Optimisation of feeder setup and component placement sequence are very important to the efficiency of surface mount placement machines. Much works have been conducted to solve this problem. However, the technological characteristics of the placement machine influences the nature of the planning prob ..."
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Cited by 13 (10 self)
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Optimisation of feeder setup and component placement sequence are very important to the efficiency of surface mount placement machines. Much works have been conducted to solve this problem. However, the technological characteristics of the placement machine influences the nature of the planning problems to be solved and the formulation of the associated models. As a result, little consensus exists as to what a suitable model should be for a given machine characteristics, and the formulations proposed by different authors tend to be difficult to compare. Hence, this paper will survey the relation between models, assembly machine technologies and heuristic methods.
An Efficient Algorithm for Minimizing a Sum of PNorms
 SIAM Journal on Optimization
, 1997
"... We study the problem of minimizing a sum of pnorms where p is a fixed real number in the interval [1; 1]. Several practical algorithms have been proposed to solve this problem. However, none of them has a known polynomial time complexity. In this paper, we transform the problem into standard conic ..."
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Cited by 13 (2 self)
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We study the problem of minimizing a sum of pnorms where p is a fixed real number in the interval [1; 1]. Several practical algorithms have been proposed to solve this problem. However, none of them has a known polynomial time complexity. In this paper, we transform the problem into standard conic form. Unlike those in most convex optimization problems, the cone for the pnorm problem is not selfdual unless p = 2. Nevertheless, we are able to construct two logarithmically homogeneous selfconcordant barrier functions for this problem. The barrier parameter of the first barrier function does not depend on p. The barrier parameter of the second barrier function increases with p. Using both barrier functions, we present a primaldual potential reduction algorithm to compute an ffloptimal solution in polynomial time that is independent of p. Computational experiences of a Matlab implementation are also reported. Key words. Shortest network, Steiner minimum trees, facilities location, po...
A Catalog of Hanan Grid Problems
 Networks
, 2000
"... We present a general rectilinear Steiner tree problem in the plane and prove that it is solvable on the Hanan grid of the input points. This result is then used to show that several variants of the ordinary rectilinear Steiner tree problem are solvable on the Hanan grid, including  but not li ..."
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Cited by 10 (2 self)
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We present a general rectilinear Steiner tree problem in the plane and prove that it is solvable on the Hanan grid of the input points. This result is then used to show that several variants of the ordinary rectilinear Steiner tree problem are solvable on the Hanan grid, including  but not limited to  Steiner trees for rectilinear (or isothetic) polygons, obstacleavoiding Steiner trees, group Steiner trees and prizecollecting Steiner trees. Also, the weighted region Steiner tree problem is shown to be solvable on the Hanan grid; this problem has natural applications in VLSI design routing. Finally, we give similar results for other rectilinear problems. 1 Introduction Assume we are given a finite set of points S in the plane. The Hanan grid H(S) of S is obtained by constructing vertical and horizontal lines through each point in S. The main motivation for studying the Hanan grid stems from the fact that it is known to contain a rectilinear Steiner minimum tree (RSMT)...